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Talk:Existential closedness
An "application" of Existential Closedness / the Nullstellensatz When originally writing the article, I tried to include the following fact as a corollary of the existential closedness of algebraically closed fields in the class of fields: If V'' is a geometrically integral variety over a field ''K (i.e., the base change of V'' to the algebraic closure of ''K is integral), then every base change of V'' is integral. The proof ended up getting a bit out of hand. I'm moving it to this talk page since it got a bit off topic for this article: As another "application", let us prove: '''Corollary:' Let K'' be an algebraically closed field, and ''L be a field extending K''. Let ''A be a K''-algebra, possibly infinite dimensional. Suppose ''A is integral. Then so is A \otimes_K L . (In algebro-geometric terms, this means that if a variety is geometrically integral, it remains integral under any base change.) Proof: We will instead prove a slightly stronger statement: let V'' and ''W be K''-vector space, and let \mu : V \otimes_K V \to W be some ''K-bilinear pairing. Tensoring everything with L'', we get : \mu_L : (V \otimes_K L) \otimes_L (V \otimes_K L) \to W \otimes_K L Suppose that \mu(v_1,v_2) \ne 0 whenever v_1 \ne 0 \ne v_2 . Then \mu_L has the same property. To see this, write V as a directed limit of finite dimensional submodules : V = \bigcup_{i \in I} V_i Let W_i \subset W be the ''K-span of \mu(V_i,V_i) , so that \mu induces a map V_i \otimes V_i \to W_i for each i \in I , and the two maps : V_i \otimes V_i \to W_i \hookrightarrow W : V_i \otimes V_i \hookrightarrow V \otimes V \to W agree. The functor - \otimes_K L is a left adjoint, so it preserves direct limits. Also, because K'' is a field, ''L is flat, so this functor preserves injections. Therefore, : V \otimes_K L = \bigcup_{i \in I} (V_i \otimes_K L) So, if there exist two non-zero vectors v_1, v_2 \in V \otimes_K L with \mu_L(v_1,v_2) = 0 , then some V_i \otimes_K L already contains v_1, v_2 . Replacing V'' with ''V''i and ''W with W''i, we may assume that ''V and W'' are finite dimensional. In fact, we may assume that they are ''K''n and ''K''m for some ''n and m''. Then \mu is described by structure coefficients: : \mu(e_i,e_j) = \sum_k c_{ijk} e_k where the ''e''i are standard basis vectors. The same structure coefficients are correct after tensoring with ''L. Our assumption means that in K'', the following system of equations cannot be solved: : \bigwedge_{k = 1}^m \sum_{i,j} c_{ijk} v^i_1 v^j_2 : \prod_{i = 1}^n v^i_1 \ne 0 : \prod_{i = 1}^n v^i_2 \ne 0 in the variables v^1_1, v^2_1, \ldots, v^n_1, v^1_2, v^2_2,\ldots, v^n_2 . By existential closedness of ''K in L'', the same system of equations cannot be solved in ''L, which is what we wanted to show. QED. Will Johnson (talk) 07:15, November 15, 2013 (UTC)